Spectral properties of n-normal operators

Abstract

For a bounded linear operator $T$ on a complex Hilbert space and $n \in \n$, $T$ is said to be $n$-normal if $T^*T^n = T^n T^*$. In this paper we show that if $T$ is a $2$-normal operator and satisfies $\sigma(T) \cap (-\sigma(T)) \subset \{ 0 \}$, then $T$ is isoloid and $\sigma(T) = \sigma_a(T)$. Under the same assumption, we show that if $z$ and $w$ are distinct eigenvalues of $T$, then $\ker(T-z) \, \bot \, \ker(T - w)$. And if non-zero number $z \in \C$ is an isolated point of $\sigma(T)$, then we show that $\ker(T - z)$ is a reducing subspace for $T$. We show that if $T$ is a $2$-normal operator satisfying  $\sigma(T) \cap (-\sigma(T)) = \emptyset$, then Weyl's theorem holds for $T$. Similarly, we show spectral properties of $n$-normal operators under similar assumption. Finally, we introduce $(n,m)$-normal operators and show some properties of this kind of operators.